A Systematic Approach for Geometrical and Dimensional Tolerancing in
4. Geometrical tolerancing in reverse engineering
ζ=1, 2, …, j 1 ≤ j ≤ 20
minRdM_h NSCAN_n+ FDCAN_q ∀ FDCAN_q∈ FDCAN (14) NSCAN_n + FDCAN_q + ITCAN_h_κ maxRdM_h ∀ FDCAN_q∈ FDCAN (15) κ=1, 2, …, i 1≤ i ≤ 20
(a) (b) Fig. 1. Suggested Basic Hole fits qualification
3.4 Sets of preferred fits
A limited number of Preferred Fits out of the Suggested ones is proposed in Step (d) through the consideration of ISO proposed fits. Moreover, the implementation of manufacturing guidelines, such as the fact that it is useful to allocate a slightly larger tolerance to the hole than the shaft, preference of Basic Hole fits over Basic Shaft ones, preference of nominal sizes that are expressed in integers or with minimum possible decimal places etc, are additionally used to “filter” the final range of the preferred fits. The final selection, Step (e), out of the limited set of preferred fits and the method end result is reached by the consideration of the machine shop capabilities and expertise in conjunction with the application of the cost – effective RE tolerancing approach, presented in Section 5 of the chapter.
tolerancing systems are not fully compatible, they both define position geometrical tolerance as the total permissible variation in the location of a feature about its exact true position. For cylindrical features such as holes or bosses the position tolerance zone is usually the diameter of the cylinder within which the axis of the feature must lie, the center of the tolerance zone being at the exact true position, Figure 2, whereas for size features such as slots or tabs, it is the total width of the tolerance zone within which the center plane of the feature must lie, the center plane of the zone being at the exact true position. The position tolerance of a feature is denoted with the size of the diameter of the cylindrical tolerance zone (or the distance between the parallel planes of the tolerance zone) in conjunction with the theoretically exact dimensions that determine the true position and their relevant datums, Figure 2. Datums are, consequently, fundamental building blocks of a positional tolerance frame in positional tolerancing. Datum features are chosen to position the toleranced feature in relation to a Cartesian system of three mutually perpendicular planes, jointly called Datum Reference Frame (DRF), and restrict its motion in relation to it.
Positional tolerances often require a three plane datum system, named as primary, secondary and tertiary datum planes. The required number of datums (1, 2 or 3) is derived by considering the degrees of freedom of the toleranced feature that need to be restricted.
Change of the datums and/or their order of precedence in the DRF results to different geometrical accuracies, (Kaisarlis et al., 2008).
Fig. 2. Cylindrical tolerance zone and geometric true position tolerancing for a cylindrical feature according to ISO 1101 (Kaisarlis et al, 2008)
The versatility and economic benefits of true position tolerances are particularly enhanced when they are assigned at the Maximum Material Condition (MMC). At MMC, an increase in position tolerance is allowed, equal to the departure of the feature from the maximum material condition size, (ISO, 1988; ASME, 2009). As a consequence, a feature with size beyond maximum material but within the dimensional tolerance zone and its axis lying inside the enlarged MMC cylinder is acceptable. The accuracy required by a position tolerance is thus relaxed through the MMC assignment and the reject rate reduced.
Moreover, according to the current ISO and ASME standards, datum features of size that are included in the DRF of position tolerances can also apply on either MMC, Regardless of Feature Size (RFS) or Least Material Condition (LMC) basis.
Position tolerances mainly concern clearance fits. They achieve the intended function of a clearance fit by means of the relative positioning and orientation of the axis of the true
geometric counterpart of the mating features with reference to one, two or three Cartesian datums. The relationship between mating features in such a clearance fit may be classified either as a fixed or a floating fastener type, (ASME, 2009; Drake, 1999), Figure 3. Floating fastener situation exists where two or more parts are assembled with fasteners such as bolts and nuts, and all parts have clearance holes for the bolts. In a fixed fastener situation one or more of the parts to be assembled have clearance holes and the mating part has restrained fasteners, such as screws in threaded holes or studs. The approach that is here presented deals with both the floating and fixed fastener cases by integrating the individual case methodologies published by (Kaisarlis et al. 2007; 2008).
Fig. 3. Typical floating and fixed fasteners and worst case assembly conditions (Drake, 1999) Basic issues of the assignment of a Position Tolerance in RE are included in Table 1. Limited number of reference components that does not allow for statistical analysis, availability or not of the mating parts and the presence of wear may affect the reliability of the RE results.
Moreover, datum selection and the size of the position tolerance itself should ensure, obviously, a stress-free mechanical mating. The analytic approach presented in this section deals with the full range of these issues in order to produce a reliable solution within realistic time.
i. The number of available RE components that will be measured. The more RE parts are measured, the more reliable will be the extracted results. Typically, the number of available RE components is extremely limited, usually ranging from less than ten to a single one article.
ii. Off the shelf, worn or damaged RE components. Off the shelf RE components are obviously ideal for the job, given that the extent of wear or damage is for the majority of cases difficult to be quantified or compensated.
iii. Accessibility of the mating part (-s).
iv. Existence of repetitive features in the RE component that may have the same function (group or pattern of features).
v. Type of assembly (e.g. floating or fixed fasteners).
vi. The size and the form (cylindrical, circular, square, other) of the geometrical tolerance zone.
vii. Candidate datums and datum reference frames. Depending on the case more possible DRFs may be considered.
viii. Precedence of datum features in DRFs.
ix. Theoretical (basic) dimensions involved.
x. Assignment of Maximum Material and Least Material Conditions to both the RE-feature and RE datum features.
xi. Measurement instrumentation capabilities in terms of final uncertainty of the measurements results. Measurements methods and software.
Table 1. Issues of Geometrical Tolerance Assignment in RE
Assignment of RE-position tolerance for both the fixed and the floating fastener case is accomplished by the present method in five sequential steps. The analysis is performed individually for each feature that has to be toleranced in the RE-component. At least two RE reference components, intact or with negligible wear, need to be available in order to minimize the risk of measuring a possibly defective or wrongly referenced RE component and, as it is later explained in this section, to improve the method efficiency. This does not certainly mean that the method cannot be used even when only one component is available.
Mating part availability is desirable as it makes easier the datum(s) recognition. Minimum assembly clearance and, as well as, the dimensional tolerance of the RE-feature (hole, peg, pin or screw shaft) and RE-Datums (for features of size) are taken as results from the RE dimensional tolerance analysis presented in the previous section of the chapter in conjunction with those quoted in relevant application- specific standards.
The primary step (a) of the analysis concerns the recognition of the critical features on the RE component that need to be toleranced and, as well as, their fastening situation. This step is performed interactively and further directs the analysis on either the fixed or the floating fastener option. In step (b) mathematical relationships that represent the geometric constraints of the problem are formulated. They are used for the establishment of an initial set of candidate position tolerances. The next step (c) qualifies suggested sets out of the group (b) that have to be in conformance with the measured data of the particular RE-feature. The step (d) of the analysis produces then a set of preferred position tolerances by filtering out the output of step (c) by means of knowledge-based rules and/or guidelines. The capabilities and expertise of the particular machine shop, where the new components will be produced, and the cost-tolerance relationship, are taken into consideration in the last step (e) of the analysis, where the required position tolerance is finally obtained. For every datum feature that can be considered for the position tolerance assignment of an RE-feature, the input for the analysis consists of (i) the measured form deviation of the datum feature (e.g. flatness), (ii) its measured size, in case that the datum is a feature of size (e.g. diameter of a hole) and (iii) the orientation deviation (e.g. perpendicularity) of the RE-feature axis of symmetry with respect to that datum. The orientation deviations of the latter with respect to the two other datums of the same DRF have also to be included (perpendicularity, parallelism, angularity). Input data relevant with the RE-feature itself include its measured size (e.g.
diameter) and coordinates, e.g. X, Y measured dimensions by a CMM, that locate its axis of symmetry. Uncertainty of the measured data should conform to the pursued accuracy level.
In that context the instrumentation used for the measured input data, e.g. ISO 10360-2 accuracy threshold for CMMs, is considered appropriate for the analysis only if its uncertainty is at six times less than the minimum assembly clearance.
4.1 Sets of candidate position tolerance sizes
The size of the total position tolerance zone is determined by the minimum clearance, minCL, of the (hole, screw-shaft) assembly. It ensures that mating features will assemble even at worst case scenario, i.e. when both parts are at MMC and located at the extreme ends of the position tolerance zone (ASME, 2009). The equations (16 -i) and (16-ii) apply for the fixed and floating fastener case respectively,
(i) TPOS = minCL =TPOS_s + TPOS_h = MMCh – MMCs
(ii) TPOS = minCL = TPOS_h (16)
For the fixed fasteners case, in industrial practice the total position tolerance TPOS of equation (16-i) is distributed between shaft and hole according to the ease of manufacturing, production restrictions and other factors that influence the manufacturing cost of the mating parts. In conformance with that practice a set of 9 candidate sizes for the position tolerance of the RE-shaft, RCAN_s and/ or the RE-hole, RCAN_h, is created by the method with a (TPOS
/10) step, which includes the 50% -50% case,
RCAN_h = RCAN_s = {TPOS1 , TPOS2 ,…, TPOSi ,…,TPOS9}
where, TPOSi = i ·TPOS /10, i=1, 2, ..., 9 (17) For the floating fasteners case the total position tolerance TPOS of equation (16-ii) actually concerns only RE-features of the Hole type. Therefore, the RCAN_h set only contains the TPOS
element. The above tolerances attain, apparently, their maximum values when the RE feature own dimensional tolerance zone is added,
TPOSi_MAX = TPOSi + LMCh – MMCh (RE-feature / Hole)
TPOSi_MAX = TPOSi + MMCs – LMCs (RE- feature / Shaft) (18) 4.2 Sets of candidate DRFs and theoretical dimensions
To ensure proper RE-part interfacing and safeguard repeatability, datum features of the original part and those of the RE-part should, ideally, coincide. In order to observe this principle the original datum features and their order of precedence have to be determined.
Initial recognition of datum features among the features of the RE-part is performed interactively following long established design criteria for locating or functional surfaces and the same, and taking into consideration the mating parts function. Out of all candidate recognized datums an initial set of candidate DRFs, DCAN_INIT, is produced by taking all combinations in couples and in triads between them. A valid DRF should conform with the constraints that have to do with the arrangement and the geometrical deviations of its datums. Only DRFs that arrest all degrees of freedom of the particular RE-feature and consequently have three or at least two datums are considered. DRF qualification for geometric feasibility is verified by reference to the list of the valid geometrical relationships between datums as given in (ASME, 1994). The geometric relationship for instance, for the usual case of three datum planes that construct a candidate DRF is in this way validated, i.e.
the primary datum not to be parallel to the secondary and the plane used as tertiary datum not to be parallel to the line constructed by the intersection of the primary and secondary datum planes. Planar or axial datum features are only considered by the method as primary when the axis of the RE-feature is perpendicular in the first case or parallel, in the second one, to them.
The following analysis applies for both the hole and the shaft and is common for the fixed and floating fasteners case. Consequently, the indexes “h” or “s” are not used hereafter. It is here also noted that the index “i” only concerns the fixed fastener case. For the floating fastener case the index “i” has a constant value of 1. Let RFDF be the set of the measured form deviations of a candidate datum feature and RO the orientation deviations of the RE feature axis of symmetry with respect to that datum. Fitness of the members of the initial DRF set, DCAN_INIT, against the members of the RCAN set of candidate position tolerance sizes is confirmed regarding the primary datum through the following constraints,
max(RFDF)≤ TPOSi (19)
max(RO)≤ TPOSi (20)
Mutual orientation deviations of the secondary and/or tertiary datums, RODF, in a valid DRF should also conform with the position tolerance of equation (16),
max(RFmax(RODF )≤ k·TDF)≤ k·TPOSi POSi , max(RO)≤ k·T, k ≥1 POSi (21) where k is a weight coefficient depending on the accuracy level of the case. A set of Candidate DRFs is thus created, DCAN (i), that is addressed to each i member (i=1,…9) of the RCAN set.
Sets of Candidate Theoretical Dimensions, [(CCAN(ij)X, CCAN(ij)Y), i=1,2,…9, j=1,2,…,n], which locate the RE feature axis of symmetry with reference to every one of the n candidate DRF(i)j
of the DCAN(i) set are generated at the next stage of the analysis. Measured, from all the available RE reference parts, axis location coordinates are at first integrated into sets, [CCAN(ij)XM, CCAN(ij)YM]. Sets of Candidate Theoretical Dimensions are then produced in successive steps starting from those calculated from the integral part of the difference between the minimum measured coordinates and the size of the position tolerance, TPOSi,
X(ij)1 = int[min(CCAN(ij)XM ) – TPOSi],Y(ij)1 = int[min(CCAN(ij)YM) – TPOSi] (22) Following members of the CCAN (ij)X, CCAN (ij)Y sets are calculated by an incremental increase
δ of the theoretical dimensions X(ij)1, Y(ij)1 ,
X(ij)2 = X(ij)1 + δ, Y(ij)2 = Y(ij)1 + δ X(ij)3 = X(ij)2 + δ, Y(ij)3 = Y(ij)2 + δ ……… ………
X(ij)p = X(ij)(p-1) + δ, Y(ij)q = Y(ij)(q-1) + δ
(23)
where as upper limit is taken that of the maximum measured X(ij)M, Y(ij)M coordinates plus the position tolerance TPOSi,
X(ij)p ≤ max(CP(ij)XM)+ TPOSi, Y(ij)q ≤ max(CP(ij)YM)+ TPOSi (24) with the populations p, q of the produced CCAN(ij)X and CCAN(ij)Y sets of candidate theoretical dimensions not necessarily equal. In the case study that is presented δ=0.05mm. Other δ- values can be used as well.
4.3 Sets of suggested position tolerances
Sets of Suggested DFRs that are produced in step (b), DSG(i), are qualified as subgroups of the sets of Candidate DFRs, DCAN(i), in accordance with their conformance with the measured location coordinates and the application or not of the Maximum or Least Material Conditions to the RE-feature size or to the RE-Datum size. In conjunction with equation (16), qualification criterion for the Suggested DFR’s, DRF(i)j j=1,2,…, n, is, Figure 4(a),
max{ΔX(ij)M, ΔΥ(ij)M} ≤ TPOSi (25)
(a) (b) Fig. 4. Qualification conditions for suggested DRFs (Kaisarlis et al., 2007) where,
ΔX(ij)M = max(CCAN(ij)XM)– min(CCAN(ij)XM)
ΔΥ(ij)M= max(CCAN (ij)YM)– min(CCAN(ij)YM) (26) In case constraint (25) is not satisfied, a DRF(i)j can only be further considered, when Maximum or Least Material Conditions are applied to RE-feature size, Figure 4(b),
max{ΔX(ij)M, ΔΥ(ij)M} ≤ TPOSi_MAX (27)
In case no member of a DCAN(i) (i=1,2,…9) set satisfies either constraint (25) or constraint (27) the relevant TPOSi is excluded from the set of Suggested Position Tolerance Sizes, RSG. Let r be the number of the available RE-parts. Sets of Suggested Theoretical Dimensions, [CSG(ij)X, CSG(ij)Y], can now be filtered out of the Candidate Theoretical Dimensions through the application of the relationships,
(a) (b)
Fig. 5. Qualification conditions for suggested theoretical dimensions (Kaisarlis et al., 2007) | X(ij)m – X(ij)Mu | ≤
2
TPOSi_MAX , |Y(ij)k – Y(ij)Mu | ≤ 2
TPOSi_MAX (28)
m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r
and the constraint imposed by the geometry of a position tolerance, Figure 5(a),
(X(ij)m – X(ij)Mu ) 2 + (Υ(ij)k– Υ(ij)Mu ) 2 ≤ POSi )2 2
(T (29)
m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r
Candidate Theoretical Dimensions that satisfy the constraints (28) but not the constraint (29) can apparently be further considered in conjunction with constraint (27) when Maximum or Least Material Conditions are used. In these cases they are respectively qualified by the conditions, e.g. for the case of RE-feature /Hole, Figure 5(b),
(X(ij)m– X(ij)Mu ) 2 + (Υ(ij)k– Υ(ij)Mu ) 2 ≤ POSi Mu )2 2
MMC - d +
(T (30)
(X(ij)m– X(ij)Mu ) 2 + (Υ(ij)k– Υ(ij)Mu ) 2 ≤ POSi Mu )2 2
d - LMC +
(T (31)
m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r
When applicable, the case of MMC or LMC on a RE-Datum feature of size may be also investigated. For that purpose, the size tolerance of the datum, TS_DF, must be added on the right part of the relationships (27) and (28). In that context, the constraints (30) and (31), e.g.
for the case of RE-feature /Hole - RE-Datum /Hole on MMC, are then formulated as, (X(ij)m– X(ij)Mu ) 2 + (Υ(ij)k– Υ(ij)Mu ) 2 ≤ POSi Mu Mu_DF DF )2
2
MMC - d + MMC - d +
(T (32)
(X(ij)m– X(ij)Mu ) 2 + (Υ(ij)k– Υ(ij)Mu ) 2 ≤ POSi Mu Mu_DF DF )2 2
MMC - d + d - LMC +
(T (33)
m=1,2, …,p ; k=1,2,…,q ; u=1,2,…,r
where dMu_DF is the measured diameter of the datum on the u-th RE-part and MMCDF the MMC size of the RE-Datum.
4.4 Sets of preferred position tolerances
The next step of the analysis provides for three tolerance selection options and the implementation of manufacturing guidelines for datum designation in order the method to propose a limited number of Preferred Position Tolerance Sets out of the Suggested ones and hence lead the final decision to a rational end result. The first tolerance selection option is only applicable in the fixed fasteners case and focuses for a maximum tolerance size of a TPOS/2. The total position tolerance TPOS, whose distribution between the mating parts is unknown, will be unlikely to be exceeded in this way and therefore, even in the most unfavourable assembly conditions interference will not occur. The second selection option gives its preference to Position Tolerance Sets that are qualified regardless of the application of the Maximum or Least Material Conditions to the RE-feature size and/ or the RE- datum feature size i.e. through their conformance only with the constraint (29) and not the constraints (30) to (33). Moreover, guidelines for datums which are used in the above context are, (ASME 2009; Fischer, 2004):
- A datum feature should be: (i) visible and easily accessible, (ii) large enough to permit location/ processing operations and (iii) geometrically accurate and offer repeatability to prevent tolerances from stacking up excessively